RE: Applicability of Maths to the Universe
June 10, 2020 at 12:22 pm
(This post was last modified: June 10, 2020 at 12:49 pm by polymath257.)
There are two sides for mathematics: math as mathematicians do it, and math as other people use it.
For a mathematician, mathematics is a formal system, defined by axioms and rules of inference. The goal, for a mathematician, is to produce 'beautiful mathematics'. But it is a characteristic of that formal system that it allows for a great many different subsystems. So, we can talk about all sorts of different geometries, all sorts of different number systems, all sorts of different combinatorial systems, etc. It allows for modeling a great number of smaller formal systems, each with its own characteristics and properties.
Then, there is math as people who want to *use* it see it. For these people, math is a language that is useful for modeling what is observed. And the fact that this language allows for a great variety of patterns means that there is almost always some aspect of mathematics that can model what is observed. Even in the worst case scenario, new math can be invented that helps to describe what is seen.
So, yes, groups were studied by mathematicians long before they were found to be useful for chemistry and physics. But, since groups are, in essence, descriptions of possible symmetries, it is hardly surprising that physical situations involving symmetry can be described by group theory. Similarly, manifolds are a type of 'smooth geometry' that allow a form of calculus to be done, so it is hardly surprising that it is found to be a useful concept when physicists started thinking that maybe the geometry of spacetime is a thing to be studied. Also, don't forget the 'ideas in the air' aspect: if mathematicians have studied it, it is far more likely that a physicist will decide to use that formalization in their own work.
And, of course, when discussing the 'unreasonable effectiveness of mathematics', it is important to acknowledge the wide swaths of mathematics that have NO known connection to any physical theories whatsoever. And, frankly, this is a very large part of mathematics. it is also important to realize that other aspects of mathematics were *invented* to help describe things in science. So it is hardly surprising that these areas are found to be useful.
And, don't forget that because math is incredibly diverse, if one model fails, there is almost always another model close by that will succeed. if flat spacetime doesn't work, use curved; if groups don't work, use monoids; if monoids don't work, use categories, etc.
Plato liked to use math as a key point in his argument that there are 'eternal forms', but I think the revelations of modern math have made that a rather untenable position. We *know* there is more than one possible geometry. We *know* there is more than one possible number system. We *know* that there are other axioms we could use and get much of the same expressiveness. We also *know* that many ideas in math will never have a precise correspondence in the physical world. For example, whether the Continuum Hypothesis is true or not will have no effect on any physical theory I can imagine, but we *know* that we can either assume it to be true OR assume it to be false *and get equally consistent mathematics*.
TL;DR: math is a sport for mathematicians, and a very expressive language for others. It is expressive enough to allow for many different ways to describe almost anything.
Burtt's book is limited by it focus (necessary at the time of publication) to Newtonian physics. But the metaphysical problems that arose from relativity and quantum mechanics go far beyond those presented by Newtonian physics. That this book was written in 1932, just after the formulation of Schrodinger's equation, is quite enough to question its applicability to modern science.
For a mathematician, mathematics is a formal system, defined by axioms and rules of inference. The goal, for a mathematician, is to produce 'beautiful mathematics'. But it is a characteristic of that formal system that it allows for a great many different subsystems. So, we can talk about all sorts of different geometries, all sorts of different number systems, all sorts of different combinatorial systems, etc. It allows for modeling a great number of smaller formal systems, each with its own characteristics and properties.
Then, there is math as people who want to *use* it see it. For these people, math is a language that is useful for modeling what is observed. And the fact that this language allows for a great variety of patterns means that there is almost always some aspect of mathematics that can model what is observed. Even in the worst case scenario, new math can be invented that helps to describe what is seen.
So, yes, groups were studied by mathematicians long before they were found to be useful for chemistry and physics. But, since groups are, in essence, descriptions of possible symmetries, it is hardly surprising that physical situations involving symmetry can be described by group theory. Similarly, manifolds are a type of 'smooth geometry' that allow a form of calculus to be done, so it is hardly surprising that it is found to be a useful concept when physicists started thinking that maybe the geometry of spacetime is a thing to be studied. Also, don't forget the 'ideas in the air' aspect: if mathematicians have studied it, it is far more likely that a physicist will decide to use that formalization in their own work.
And, of course, when discussing the 'unreasonable effectiveness of mathematics', it is important to acknowledge the wide swaths of mathematics that have NO known connection to any physical theories whatsoever. And, frankly, this is a very large part of mathematics. it is also important to realize that other aspects of mathematics were *invented* to help describe things in science. So it is hardly surprising that these areas are found to be useful.
And, don't forget that because math is incredibly diverse, if one model fails, there is almost always another model close by that will succeed. if flat spacetime doesn't work, use curved; if groups don't work, use monoids; if monoids don't work, use categories, etc.
Plato liked to use math as a key point in his argument that there are 'eternal forms', but I think the revelations of modern math have made that a rather untenable position. We *know* there is more than one possible geometry. We *know* there is more than one possible number system. We *know* that there are other axioms we could use and get much of the same expressiveness. We also *know* that many ideas in math will never have a precise correspondence in the physical world. For example, whether the Continuum Hypothesis is true or not will have no effect on any physical theory I can imagine, but we *know* that we can either assume it to be true OR assume it to be false *and get equally consistent mathematics*.
TL;DR: math is a sport for mathematicians, and a very expressive language for others. It is expressive enough to allow for many different ways to describe almost anything.
(June 10, 2020 at 8:33 am)Belacqua Wrote:(June 10, 2020 at 7:59 am)Grandizer Wrote: It's Penrose, so I'm going to take him seriously.
He's a smart guy!
Here's an article I found just now, by coincidence. It touches on a number of issues related to the topic here, I think. It's particularly interesting to me how Newton changed what science was looking for, by giving up on material causal explanations and looking instead for mathematization. So in a way, science says it has understood the material when it stops looking at the material and changes it into an abstraction -- math.
https://americanaffairsjournal.org/2020/...t-meaning/
Much of what the writer refers to in passing is described in detail in Burtt's classic book The Metaphysical Foundations of Modern Science.
https://www.amazon.com/Metaphysical-Foun...oks&sr=1-1
Which may be pirated here:
http://gen.lib.rus.ec/book/index.php?md5...7F3D3AD067
As an early critic of the Enlightenment, William Blake accused Newton of valuing the abstracted understanding over the genuine material -- of claiming that truth is in the formula rather than in the thing itself.
Burtt's book is limited by it focus (necessary at the time of publication) to Newtonian physics. But the metaphysical problems that arose from relativity and quantum mechanics go far beyond those presented by Newtonian physics. That this book was written in 1932, just after the formulation of Schrodinger's equation, is quite enough to question its applicability to modern science.
